The method for carrying Essay Example for Free

The method for carrying Essay My aim is to investigate the mathematical propagation of bad tomatoes This is essentially an investigation of patterns derived from a simple set of rules for this propagation, in the manner of a simplified life genesis program. The rules are as followed: 1. The first hour, any one of the tomatoes (depending on the investigation) turns bad 2. From that hour on, any tomato touched by a bad tomato will turn bad itself, on an hourly basis. 3. Tomatoes are constrained within an nn grid, which restricts propagation of bad tomatoes. As visible from the rules, this allows for creation of simple models to show the propagation of bad tomatoes. From these, I hope to derive formulae, or sets of rules if formulae are not possible, to make logical predictions. We shall define the variables as will be used in the description of this investigation as follows: n The hour in which a tomato turns g The grid size (g2) x The number of turned tomatoes in each n h The number of hours taken for all tomatoes to go bad t Total number of turned tomatoes (equal to g2) Contents Item Page number Introduction 1 Contents 2 Mapping of tomatoes in the middle of a side 3 Tomatoes in the corner 7 Conclusion. This grid represents the propagation of bad tomatoes in an nxn square, covering grids up to size 2424. Some of the results for this data are plotted on the table below: While at first it seems the patterns in this table should be obvious, this is deceptive. Only by splitting the table into three regions do we see the separate patterns defining the table. These regions, as shown in the following table, allow patterns to emerge. These patterns do not, as you would expect, work down with different numbers in the same grids, but instead work across with the same number in different grids. In the first region (yellow), we see that, in every case, x is equal to n+n-1. The latter two regions (green and purple) are substantially harder, and require a sequential approach. Naturally, the first step in devising a formula, to take n and g and return x, is determining which region the number lies in. This is a simple matter of comparing g with n. Once we know the region, we can use a set op steps to calculate the number x. The method for carrying out this operation will be described shortly. The left grid shows an updated version of my results demonstrating the three regions yellow, green and purple, as well as some extra data formulated from the patterns observed. This is the first step in trying to formulate equations to work on all situations. Before moving on to the main essence of the project, finding a formula to derive x from n and g, we shall examine a few other formulae not directly related to this but still relevant to the investigation. Â  To find the total number of hours taken for all tomatoes to go bad within a grid, you use a formula depending on g. This formula also depends on whether g is odd or even: Â  If g is odd, then h=((g+1)/2)-1 If g is even then h=(g/2)+1 Â  In all square grid situations, t is always g2. The number of tomatoes to turn each hour in an infinite grid, starting on the side in the centre is equal to 2n-1. The total number of tomatoes that are bad after each hour is equal to n2. We shall briefly describe the patterns used to expand this table and in the following formulae: Yellow numbers always go up by 0 each grid size Green numbers go up by 1 Purple numbers go up by 3 Green/yellow boundaries go up by 1 Purple/green boundaries go up by 2 We now move on to analyse the main problem: the individual number of tomatoes to turn in each hour. This, as mentioned earlier, is a much more complicated program, and requires division of the grid into three regions. The following steps attempt to demonstrate how, and why, this is done. 1. The first step is to compare n with g, to work out which region the answer is likely to lie in. For this example we shall use two numbers, grid size 24 and tomato number 25. Compare n with g: If ng, x lies in the purple region If n=g, x lies in the green region If gn, x lies in green or yellow and further calculation is needed: If g is odd: if g= n-((g-1)/2) x is yellow, and if gn-((g-1)/2) then x is green If g is even: if g= n-(g/2) x is yellow, and if gn-(g/2) x is green We then move to region specific instructions: Yellow x =2n-1 Green x =g Purple. (Calculating purple numbers is substantially more complex) (Also note the existence of bln, a new variable we introduce here whose meaning will be explained later) Do n mod 3: N mod 3 = 0 then Bln = 2(n/3) N mod 3 = 1 then Bln = (2((n+2)/3))-1 N mod 3 = 2 then Bln = 2((n+1)/3) Do g bln Again, look at n mod 3: If 0, multiply last number by 3 and add 1 If 1, multiply last number by 3 and add 2 If 2, multiply last number by 3 and add 3 Therefore, by this process we can calculate any number from the grid size and the hour. For our example, g = 24 and n = 25, we would do the following: 1.n g, therefore x is purple 2. 25 mod 3 is 1, therefore bln = 2(27/3))-1 = 17 3. 24 17 is 7 4. 25 mod 3 is 1, therefore we: 5. Multiply 7 by 3 = 21 6. And add 2, giving 23 I have checked this with both an extended table of results (created using the patterns found earlier), and with a small excel macro designed to count the numbers of tomatoes turned each hour. Both yield the same result. Â  The left is the segment from my expanded table showing the result. The 23 in the middle of the table represents grid size 24 and hour 25 what my formula predicted. The left here is the automatic count from my macro. The data reads (for a 2424 table) hour,count (or n,x). This also agrees with my prediction. We shall here briefly explain how the purple formula works (formulas for both green and yellow are self-explanatory). I observed that the base line (the line marking the bottom of the purple section- representative of the number of tomatoes to turn bad in the final hour) of the purple section follows a three stage recurring pattern. Because we are working from the base line to reach our result, as the numbers go up by 3 each time, calculating the start point and value of the base line for each hour was essential. To work easily with a three -stage recurrence, we needed to work in base 3, the easiest implementation of which involves modulo arithmetic. By doing n mod 3, we work out which stage of the cycle represents the first grid size for tomatoes to turn in a particular hour. Once the cycle is split, we can show different formulae for each stage, derived from observance of the patterns.

Ethics Paper -- essays research papers

Everyday we each face questions of what we ought to do. We sometimes ask ourselves, â€Å"What if everyone did that?† Every time you decide to pick up a piece of trash because you want the city to look nice, you are not doing it because of the aesthetic effect of one piece of trash, but rather what the city would look like if no one picked up their trash. Kant uses this everyday question in his system of morality as part of the categorical imperative. For Kant, the morality of an action can be determined by the categorical imperative. Kant would like to determine the morality of stealing, therefore Kant wants to examine the morality of â€Å"I will steal anything I want to satisfy my desire for it†. Then Kant rephrases the statement to ask the question of what if everybody did it, â€Å"Everyone will steal anything they want to satisfy their desire for it.† Then Kant makes that statement a maxim, a law which must be followed by everyone in Kant’s test world. Kant examines the world and asks if you can consistently will your maxim in a world in which that is a law? But if everyone steals anything they desire, how will there be property rights since it is okay for anyone to take anything at any time? There can’t. Since there are no property rights, the maxim breaks down since stealing only occurs when someone takes property from its rightful owner. Since there is a contradiction in the 1 conception of the maxim, you are prohibited from acting on that maxim. Imagine Ice Man, a cold, rational person that does not find inner satisfaction in spreading joy and cannot take delight in the satisfaction of others. Does Ice Man have a duty to help others when they are in need? Ice Man is wealthy and not in need of help from others? Ice Man wants to determine the morality of â€Å"I will not help others when they are in need of help.† Therefore, what if everyone did not help others when they are in need of help. Despite this being an unhappy world, there is no contradiction in conception in this maxim unlike above. But does it pass Kant’s contradiction in willing test? Ice Man is defined as a rational being. As a rational being, Ice Man knows that one day he too will be in need. Since he is a rational being, he will prefer that someone would help him and as a rational being, cannot will that no one would help other when they are in need. Since it fails the contradiction in willing test, ev... ...by universalizing the situation and removing your own self interest, then we judge the consequences to our actions without prejudice or preference. By ignoring the question of â€Å"What if everyone did that?†, we can 3 justify murder, lies, and other unmoral acts that can hurt much more than we realize in the heat of the moment. If I leave a piece of trash on the ground when I am in rush or otherwise pressured and believe that is okay, that piece of trash on the ground means that someone else will have to pick it up and that other people will feel more free to drop their trash there.1 Many actions may seem to hurt no one, but in the aggregate do cause pain to others. By ignoring the question of â€Å"What if everyone did that?†, we ignore the infinitesimal effects our actions have on everyone we come into contact with it or simply feel the secondary effects. Therefore, the universal question of â€Å"What if everyone did that?† should be a part of our ethical thinking. 1I did an informal study on this in my social room last year with both dirty dishes in the sink and paper towels on the floor. The difference in dirtiness after an hour was impressive. Real studies have been done on this as well.

Homework Essay

Chapter 18 p534 1.What is the key assumption of the basic Keynesian model? Explain why this assumption is needed if one is to accept the view that aggregate spending is a driving force behind short-term economic fluctuations. The Keynesian model shows how fluctuations in planned aggregate expenditure can cause actual output to differ from potential output. This method is necessary because if it were not used companies would have to change prices every time there was a possible change in demand or quantity shift in inventory. With this method short term economic flux can happen when the a company does shift their price to meet demand. 3. Define planned aggregate expenditure and list its components. Why does planned spending change when output changes relatively infrequently. What accounts for the difference? This is a total planned spending on goods and services including; consumption, investment, government purchases and net ports. If spending change happens infrequently then added goods go into inventory causing company to spend capital on invested inventory. Consumption function accounts for the difference between changes in expenditure. Chapter 19 1. Why does the real interest rate affect planned aggregate expenditure? Give examples. Because the raising or lowering affects the cost of borrowing, which affects consumption and planned investment (which all is a part of aggregate expenditure). If the Fed raises rates the housing market will slow down buying. If the Fed lowers rates more people are likely to buy homes and refinance. 2. The Fed faces a recessionary gap. How would you expect it to respond? Explain step by step how its policy change is likely to affect the economy. The Fed’s position is to eliminate output gaps and maintain low inflation. To eliminate a recessionary output gap, the Fed will raise the real interest rate.

Why Is Pump Lift Limited to a Specific Height

Pump lift is the linear vertical measurement that indicates the distance a certain pump can draw a liquid from the intake into the pump body. It is then exposed to the moving parts which will compress the liquid and eject it through the outlet side of the pump. An Example For example; a pump fitted to the top of a tank must be able to perform in the most challenging conditions. In the case of the tank, thats when its almost empty. A mostly full tank is easier for the pump to draw from since the liquid in the tank will seek the same level in the intake pipe. In a mostly empty tank, the pump will have to draw liquid up the full height of the pump intake pipe. The Physical Properties The physical properties of materials like viscosity and density can impact the lift performance. Because oil is less dense than water the lift will be greater because of the ratio of weight to volume. Less weight is being lifted by the vacuum the pump creates in the inlet so a less dense material can travel higher with less energy than a denser liquid like water. The reason a pump cannot deliver fluid to the pump body has to do with the interaction of different liquids with the partial vacuum that the pump is creating in the inlet. An Experiment In an experimental display, we would be able to see containers of liquid of various densities. Each container would include a clear vertical tube which has had all matter pumped out (actually impossible) to create a perfect vacuum. We would see liquids drawn up to a certain height by the pull of the vacuum but gravity would also be pulling the liquid down Since no pump produces a perfect vacuum in the inlet the maximum pump lift of the same liquids in a real world situation would be reduced because of the inherent inefficiency of the pump mechanism. The Pump Type A more efficient pump design can use several techniques to improve lift performance. The pump type has much to do with performance. A piston type pump will always be more efficient than a centrifugal pump since its a closed chamber design. In addition to making a closed chamber design, the number of cycles per minute can be increased to allow for the lower capacity of this type of pump. Sealing the moving parts like a piston or impeller against the pump chamber can help prevent leakage and improve efficiency. Often, the easiest solution is to lower the pump or submerge it in the liquid which is sometimes not practical because of maintenance issues.